18.06 Problem Set 1 Solutions Total: 100 points
Section 1.2. Problem 23: The figure shows that cos(α) = v1 /�v� and sin(α) = v2 /�v�. Similarly cos(β) is and sin(β) is . The angle θ is β − α. Substitute into the trigonometry formula cos(α) cos(β) + sin(β) sin(α) for cos(β − α) to find cos(θ) = v · w/�v��w�. Solution (4 points) First blank: w1 /�w�. Second blank: w2 /�w�. Substituting into the trigonome try formula yields cos(β − α) = (w1 /�w�)(v1 /�v�) + (w2 /�w�)(v2 /�v�) = v · w/�v��w�.
Section 1.2. Problem 28: Can three vectors in the xy plane have u · v < 0 and v · w < 0 and u · w < 0? Solution (12 points) √ √ Yes. For instance take u = (1, 0), v = (− 12 , 23 ), w = (− 12 , − 23 ). Notice u · v = v · w = u · w = − 12 . Section 1.3. Problem 4: Find a combination x1 w1 + x2 w2 + x3 w3 that gives the zero vector: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 4 7 w 1 = ⎣ 2 ⎦ ; w2 = ⎣ 5 ⎦ ; w3 = ⎣ 8 ⎦ . 3 6 9 Those vectors are (independent)(dependent). The three vectors lie in a The matrix W with those columns is not invertible.
.
Solution (4 points) Observe w1 − 2w2 + w3 = 0. The vectors are dependent. They lie in a plane. Section 1.3. Problem 13: The very last words say that the 5 by 5 centered difference matrix is not invertible. Write down the 5 equations Cx = b. Find a 1
combination of left sides that gives zero. What combination of b1 , b2 , b3 , b4 , b5 must be zero? Solution (12 points) The 5 by 5 centered difference matrix is ⎛ 0 1 0 0 ⎜−1 0 1 0 ⎜ 1 C = ⎜ ⎜ 0 −1 0 ⎝ 0 0 −1 0 0 0 0 −1
⎞ 0
0 ⎟ ⎟ 0 ⎟ ⎟ .
1⎠
0
The five equations Cx = b are x2 = b1 , −x1 + x3 = b2 , −x2 + x4 = b3 , −x3 + x5 = b4 , −x4 = b5 . Observe that the sum of the first, third, and fifth equations is zero. Similarly, b1 + b3 + b5 = 0. Section 2.1. Problem 29: Start with the vector u0 = (1, 0). Multiply again and again by the same “Markov matrix” A = [.8 .3; .2 .7]. The next three vectors are u1 , u2 , u3 : �
� � � � �
.8 .3 1 .8 u1 = = u2 = Au1 = u3 = Au2 = . .2 .7 0 .2 What property do you notice for all four vectors u0 , u1 , u2 , u3 . Solution (4 points) Computing, we get
� u2 =
.7 .3
�
� u3 =
.65 .35
� .
All four vectors have components that sum to one. Section 2.1. Problem 30: Continue Problem 29 from u0 = (1, 0) to u7 , and also from v0 = (0, 1) to v7 . What do you notice about u7 and v7 ? Here are two MATLAB codes, with while and for. They plot u0 to u7 and v0 to v7 . The u’s and the v’s are approaching a steady state vector s. Guess that vector and check that As = s. If you start with s, then you stay with s. 2
Solution (12 points) Here is the code entered into MATLAB.
u= [1; 0]; A= [.8 .3; .2 .7]; x= u;�� k= [0 : 7];�� while size (x, 2)<=7�� u= A*u; x= [x, u];�� end�� plot (k,x)�� v= [0;1];�� A= [.8 .3; .2 .7];�� x= v;� k= [0:7];�� for j=1:7�� v= A*v; x=[x, v];�� end�� plot (k, x)��
In this graph, we see that the sequence u1 , u2 , . . . , u7 is approaching (.6, .4).
3
In this graph, we see that the sequence v1 , v2 , . . . , v7 is approaching (.6, .4).
4
From the graphs, we guess that s = (.6, .4) is a steady state vector. We verify this with the computation � �� � � � .8 .3 .6 .6 As = = . .2 .7 .4 .4
Section 2.2. Problem 20: Three planes can fail to have an intersection point,
even if no planes are parallel. The system is singular if row 3 of A is a
of the first two rows. Find a third equation that can’t be solved together with
x + y + z = 0 and x − 2y − z = 1.
Solution (4 points) The system is singular if row 3 of A is a linear combination of the first two rows. There are many possible choices of a third equation that cannot be solved together with the ones given. An example is 2x + 5y + 4z = 1. Note that the left hand side of the third equation is the three times the left hand side of thte first minus the left hand side of the second. However, the right hand side does not satisfy this relation. Section 2.2. Problem 32: Start with 100 equations Ax = 0 for 100 unknowns x = (x1 , . . . , x100 ). Suppose elimination reduces the 100th equation to 0 = 0, so the system is “singular”. (a) Elimination takes linear combinations of the rows. So this singular system has the singular property: Some linear combination of the 100 rows is . (b) Singular systems Ax = 0 have infinitely many solutions. This means that some linear combination of the 100 columns is . (c) Invent a 100 by 100 singular matrix with no zero entries. (d) For your matrix, describe in words the row picture and the column picture of Ax = 0. Not necessary to draw 100-dimensional space. Solution (12 points) (a) Zero. (b) Zero. (c) There are many possible answers. For instance, the matrix for which every row is (1 2 3 · · · 100). (d) The row picture is 100 copies of the hyperplane in 100-space defined by the equation x1 + 2x2 + 3x3 + · · · + 100x100 = 0. The column picture is the 100 vectors proportional to (1 1 1 · · · 1) of lengths 10, 20, . . . , 1000. 5
Section 2.3. Problem � 22: The entries of A and x are aij and xj . So the first component of Ax is a1j xj = a11 x1 + · · · + a1n xn . If E21 subtracts row 1 from row 2, write a formula for (a) the third component of Ax (b) the (2, 1) entry of E21 A (c) the (2, 1) entry of E21 (E21 A) (d) the first component of E21 Ax. Solution (4 points)
� �
(a) a3j xj . (b) a21 − a11 . (c) a21 − 2a11 . (d) a1j xj . Section 2.3. Problem 29: Find the triangular matrix ” to a smaller Pascal: ⎡ ⎤ ⎡ 1 0 0 0 1 0 ⎢1 1 0 0 ⎥ ⎢0 1 ⎥ ⎢ E ⎢ ⎣ 1 2 1 0 ⎦ = ⎣ 0 1 1 3 3 1 0 1
matrix E that reduces “Pascal’s 0 0 1 2
⎤ 0
0 ⎥ ⎥ .
0 ⎦
1
Which matrix M (multiplying several E’s) reduces Pascal all the way to I?
Solution (12 points)
⎡
1 0 0 ⎢−1 1 0 E = ⎢ ⎣ 0 −1 1 0 0 −1 One can eliminate the second column with the ⎡ 1 0 0 ⎢0 1 0 ⎢ ⎣ 0 −1 1 0 0 −1 and the third column with the matrix
⎡ 1 0 0 ⎢0 1 0 ⎢ ⎣ 0 0 1 0 0 −1
6
⎤ 0
0⎥ ⎥ .
0⎦
1
matrix
⎤ 0 0⎥ ⎥ 0⎦
1
⎤ 0
0 ⎥ ⎥ .
0⎦
1
Multiplying these ⎡ 1 0 0 ⎢0 1 0 M = ⎢ ⎣ 0 0 1 0 0 −1
together, we ⎤⎡ 0 1 0 ⎢ 0 ⎥ ⎥ ⎢0 1 0 ⎦ ⎣0 −1 1 0 0
get
0 0 1 −1
⎤⎡ 0 1 0 0 ⎥ ⎢ 0 ⎥ ⎢−1 1 0 0⎦ ⎣ 0 −1 1 1 0 0 −1
⎤ ⎡ 0 1 ⎥ ⎢ 0⎥ ⎢−1 = 0⎦ ⎣ 1 1 −1
0 0 1 0 −2 1 3 −3
⎤ 0 0 ⎥ ⎥ .
0 ⎦ 1
Section 2.4. Problem 32: Suppose you solve Ax = b for three special right sides
b:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0
Ax1 = ⎣ 0 ⎦ ; Ax2 = ⎣ 1 ⎦ ; Ax3 = ⎣ 0 ⎦ .
0 0 1 If the three solutions x1 , x2 , x3 are the columns of a matrix X, what is A times X? Solution (4 points) The matrix AX has columns Ax1 , Ax2 , and Ax3 . Therefore, AX = I. Section 2.4. Problem 36: Suppose A is m by n, B is n by p, and C is p by q. Then the multiplication count for (AB)C is mnp + mpq. The multiplication count from A times BC with mnq + npq separate multiplications. (a) If A is 2 by 4, B is 4 by 7, and C is 7 by 10, do you prefer (AB)C or A(BC)? (b) With N -component vectors, would you choose (uT v)wT or uT (vwT )? (c) Divide by mnpq to show that (AB)C is faster when n−1 + q −1 < m−1 + p−1 . Solution (12 points) (a) Note that (AB)C has 2 · 4 · 7 + 2 · 7 · 10 = 196 multiplications and A(BC) has 2 · 4 · 10 + 4 · 7 · 10 = 360. Hence, we prefer (AB)C. (b) We prefer (uT v)wT ; it requires 2N multiplications. On the other hand, the multiplication count for uT (vwT ) is 2N 2 . (c) Note (AB)C is faster when mnp + mpq < mnq + npq. Dividing by mnpq, we get q −1 + n−1 < p−1 + m−1 . Section 2.5. Problem 7: If A has row 1 + row 2 = row 3, show that A is not invertible: (a) Explain why Ax = (1, 0, 0) cannot have a solution. (b) Which right sides (b1 , b2 , b3 ) might allow a solution to Ax = b? (c) What happens to row 3 in elimination? 7
Solution (4 points) (a) Suppose A has row vectors A1 , A2 , A3 , and x is a solution to Ax = (1, 0, 0). Then A1 · x = 1, A2 · x = 0, and A3 · x = 0. But A1 + A2 = A3 means that A1 · x + A2 · x = A3 · x and 1 + 0 = 0, a contradiction. (b) If Ax = (b1 , b2 , b3 ), then A1 · x = b1 , A2 · x = b2 , A3 · x = b3 . Since A1 + A2 = A3 , we deduce b1 + b2 = b3 . (c) In the eliminated matrix, the third row will be zero.
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18.06 Linear Algebra Spring 2010
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